Fluids Applications of Fluid Dynamics Lana Sheridan De Anza College April 16, 2018
Last time fluid dynamics the continuity equation Bernoulli s equation
Overview Torricelli s law applications of Bernoulli s equation
Bernoulli s Equation and the Continuity Equation A law discovered by the 18th-century Swiss scientist, Daniel Bernoulli. Bernoulli s Principle As the speed of a fluid s flow increases, the pressure in the fluid decreases. The Continuity equation: A 1 v 1 = A 2 v 2 Bernoulli s Equation: P + 1 2 ρv 2 + ρgh = const is constant along a streamline in the fluid.
Torricelli s Law from Bernoulli s Equation Bernoulli s equation can also be used to predict the velocity of streams of water from holes in a container at different depths.
Torricelli s Law from Bernoulli s Equation The liquid at point 2 is at rest, at a height y 2 and pressure P. At point 1 is leaves with a velocity v 1, at a height y 1 and pressure P 0. blow more strongly and watch the increased pressure difin its side at a distance he atmosphere, and its air above the liquid is id as it leaves the hole er. When the hole is pressure P at the top speed. If the pressure extinguisher must be 1 Point 2 is the surface of the liquid. P y 2 h A 2 A 1 Point 1 is the exit point of the hole. S v 1 P y 0 1 Figure 14.20 (Example 14.9) A liquid leaves a hole in a tank at speed v 1. ρv2 2 + ρgy 2 + P 2 ρv 2 1 + ρgy 1 + P 0 = 1 2 0
Torricelli s Law from Bernoulli s Equation in its side at a distance he atmosphere, and its air above the liquid is id as it leaves the hole Point 2 is the surface of the liquid. P h A 2 A 1 Point 1 is the exit point of the hole. er. When the hole is pressure P at the top speed. If the pressure extinguisher must be y 2 S v 1 P y 0 1 Figure 14.20 (Example 14.9) 1A liquid leaves a hole in a tank at 2speed ρv 1 2 + ρgy v 1. 1 + P 0 = ρgy 2 + P Rearranging, and using y = h 2 h 1, 2(P P 0 ) v 1 = + 2gh ρ
Torricelli s Law from Bernoulli s Equation Notice that if the container is open to the air (P = P 0 ), then the speed of each jet is v = 2gh where h is the depth of the hole below the surface.
Implications of Bernoulli s Principle Bernoulli s principle also explains why in a tornado, hurricane, or other extreme weather with high speed winds, windows blow outward on closed buildings.
Implications of Bernoulli s Principle Bernoulli s principle also explains why in a tornado, hurricane, or other extreme weather with high speed winds, windows blow outward on closed buildings. The high windspeed outside the building corresponds to low pressure. The pressure inside remains higher, and the pressure difference can break the windows. It can also blow off the roof! It makes sense to allow air a bit of air to flow in or out of a building in extreme weather, so that the pressure equalizes.
Implications of Bernoulli s Principle Bernoulli s principle can help explain why airplanes can fly. Air travels faster over the top of the wing, reducing pressure there. That means the air beneath the wing pushes upward on the wing more strongly than the air on the top of the wing pushes down. This is called lift. 1 Diagram from HyperPhysics.
Air Flow over a Wing In fact, the air flows over the wing much faster than under it: not just because it travels a longer distance than over the top. This is the result of circulation of air around the wing. 1 Diagram by John S. Denker, av8n.com.
Air Flow over the Top of the Wing: Bound Vortex A starting vortex trails the wing. The bound vortex appears over the wing. Those two vortices counter rotate because angular momentum is conserved. The bound vortex is important to establish the high velocity of the air over the top of the wing. 1 Image by Ludwig Prandtl, 1934, using water channel & aluminum particles.
Wingtip Vortecies Other vortices also form at the ends of the wingtips. 1 Photo by NASA Langley Research Center.
Vortices around an Airplane 1 Diagram by John S. Denker, av8n.com.
Airflow at different Angles of Attack 1 Diagram by John S. Denker, av8n.com.
Implications of Bernoulli s Principle A stall occurs when turbulence behind the wing leads to a sudden loss of lift. The streamlines over the wing detach from the wing surface. This happens when the plane climbs too rapidly and can be dangerous. 1 Photo by user Jaganath, Wikipedia.
Implications of Bernoulli s Principle Spoilers on cars reduce lift and promote laminar flow. 1 Photo from http://oppositelock.kinja.com.
Implications of Bernoulli s Principle Wings on racing cars are inverted airfoils that produce downforce at the expense of increased drag. This downforce increases the maximum possible static friction force turns can be taken at higher speed. 1 Photo from http://oppositelock.kinja.com.
Implications of Bernoulli s Principle A curveball pitch in baseball also makes use of Bernoulli s principle. The ball rotates as it moves through the air. Its rotation pulls the air around the ball, so the air moving over one side of the ball moves faster. This causes the ball to deviate from a parabolic trajectory. 1 Diagram by user Gang65, Wikipedia.
Summary Torricelli s law applications of Bernoulli s equation Test! tomorrow, in class. (Uncollected) Homework Serway & Jewett: study for the test